Let $ABC$ be a triangle and $D$ be a point on the segment $BC$ such that $DC = 2BD$. Let $E$ be the mid-point of $AC$. Let $AD$ and $BE$ intersect in $P$. Determine the ratios $BP:PE$ and $AP:PD$.

Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7\}$.

Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$ respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$ such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove that $XY = (b + c-a)/2$ where $BC = a, CA = b $and $AB = c$.

Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.

Let ABCD be a unit square. Draw a quadrant of a circle with A as centre and B;D as end points of the arc. Similarly, draw a quadrant of a circle with B as centre and A;C as end points of the arc. Inscribe a circle ? touching the arc AC internally, the arc BD internally and also touching the side AB. Find the radius of the circle ?.

Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

Let $X=\{1,2,3,...,12\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{2,3,5,7,8\}$.

Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$.

Show that for all real numbers $x,y,z$ such that $x + y + z = 0$ and $xy + yz + zx = -3$, the expression $x^3y + y^3z + z^3x$ is a constant.

Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $ \Gamma$ touching arcs $AC$ and $BD$ both externally and also touching the side $CD$. Find the radius of $ \Gamma$.

Let $a,b,c$ be positive integers such that $a|b^5, b|c^5$ and $c|a^5$. Prove that $abc|(a+b+c)^{31}$.

Let $X=\{1,2,3,...,10\}$. Find the number of pairs of $\{A,B\}$ such that $A\subseteq X, B\subseteq X, A\ne B$ and $A\cap B=\{5,7,8\}$.

Let $ABC$ be a triangle. Let $D,E$ be points on the segment $BC$ such that $BD=DE=EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine the ratio of the area of the triangle $APQ$ to that of the quadrilateral $PDEQ$.

Find all positive integers such that $3^{2n}+3n^2+7$ is a perfect square.

Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $AC$ externally, the arc $BD$ externally and also touching the side $AD$. Find the radius of $\Gamma$.

Let $a,b,c$ be positive integers such that $a|b^2, b|c^2$ and $c|a^2$. Prove that $abc|(a+b+c)^{7}$

Let $ABC$ be a triangle. Let $E$ be a point on the segment $BC$ such that $BE = 2EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AE$ in $Q$. Determine $BQ:QF$.

Solve the system of equations for positive real numbers: $$\frac{1}{xy}=\frac{x}{z}+ 1,\frac{1}{yz} = \frac{y}{x} + 1, \frac{1}{zx} =\frac{z}{y}+ 1$$

Find with proof all non–zero real numbers $a$ and $b$ such that the three different polynomials $x^2 + ax + b, x^2 + x + ab$ and $ax^2 + x + b$ have exactly one common root.

Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.

Solve for real $x$ : $2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}$ (For a real number $x, [x]$ denotes the greatest integer less than or equal to x. For instance, $[2.5] = 2$, $[-3.1] = -4$, $[\pi ] = 3$. For a real number $x, \{x\}$ is defined as $x - [x]$.)

$H$ is the orthocentre of an acute–angled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a right–angled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$. We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$. Define WSUM for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms and the second the even numbered terms. (For example, if $A = \{2, 5, 7, 8\}$, WSUM is $3(2 + 7) + 2(5 + 8)$.) Find the sum of WSUMs over all the subsets of S. (Assume that WSUM for the null set is $0$.)

On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.

Let $x, y, z$ be positive real numbers such that $2(xy + yz + zx) = xyz$. Prove that $\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}$

Let $ABCD$ be a convex quadrilateral such that $\angle ADC=\angle BCD>90^{\circ}$. Let $E$ be the point of intersection of $AC$ and the line through $B$ parallel to $AD;$ let $F$ be the point of intersection of $BD$ and the line through $A$ parallel to $BC.$ Prove that $EF\parallel CD.$

Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$

Find all natural numbers $x,y,z$ such that \[(2^x-1)(2^y-1)=2^{2^z}+1.\]

Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.

Let $AL$ and $BK$ be the angle bisectors in a non-isosceles triangle $ABC,$ where $L$ lies on $BC$ and $K$ lies on $AC.$ The perpendicular bisector of $BK$ intersects the line $AL$ at $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK.$ Prove that $LN=NA.$

A computer program generated $175$ positive integers at random, none of which had a prime divisor grater than $10.$ Prove that there are three numbers among them whose product is the cube of an integer.